Characteristic functions as bounded multipliers on anisotropic spaces

Baladi, Viviane

doi:10.1090/proc/14107

Characteristic functions as bounded multipliers on anisotropic spaces
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Proc. Amer. Math. Soc. 146 (2018), 4405-4420 Request permission

Abstract:

We show that characteristic functions of domains with piecewise $C^3$ boundaries transversal to suitable cones are bounded multipliers on a recently introduced scale $\mathcal {U}^{\mathbf {C}, t,s}_p$ of anisotropic Banach spaces, under the conditions $-1+1/p<s<-t<0$, with $p\in (1,\infty )$.

References

V. Baladi, Dynamical Zeta Functions and Dynamical Determinants for Hyperbolic Maps, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, A series of Modern Surveys in Mathematics, Vol 68 (2018), Springer Ergebnisse.
Viviane Baladi, The quest for the ultimate anisotropic Banach space, J. Stat. Phys. 166 (2017), no. 3-4, 525–557. MR 3607580, DOI 10.1007/s10955-016-1663-0
Viviane Baladi, Mark F. Demers, and Carlangelo Liverani, Exponential decay of correlations for finite horizon Sinai billiard flows, Invent. Math. 211 (2018), no. 1, 39–177. MR 3742756, DOI 10.1007/s00222-017-0745-1
Viviane Baladi and Sébastien Gouëzel, Banach spaces for piecewise cone-hyperbolic maps, J. Mod. Dyn. 4 (2010), no. 1, 91–137. MR 2643889, DOI 10.3934/jmd.2010.4.91
Viviane Baladi and Masato Tsujii, Anisotropic Hölder and Sobolev spaces for hyperbolic diffeomorphisms, Ann. Inst. Fourier (Grenoble) 57 (2007), no. 1, 127–154 (English, with English and French summaries). MR 2313087, DOI 10.5802/aif.2253
Viviane Baladi and Masato Tsujii, Dynamical determinants and spectrum for hyperbolic diffeomorphisms, Geometric and probabilistic structures in dynamics, Contemp. Math., vol. 469, Amer. Math. Soc., Providence, RI, 2008, pp. 29–68. MR 2478465, DOI 10.1090/conm/469/09160
Michael Blank, Gerhard Keller, and Carlangelo Liverani, Ruelle-Perron-Frobenius spectrum for Anosov maps, Nonlinearity 15 (2002), no. 6, 1905–1973. MR 1938476, DOI 10.1088/0951-7715/15/6/309
Mark F. Demers and Carlangelo Liverani, Stability of statistical properties in two-dimensional piecewise hyperbolic maps, Trans. Amer. Math. Soc. 360 (2008), no. 9, 4777–4814. MR 2403704, DOI 10.1090/S0002-9947-08-04464-4
Mark F. Demers and Hong-Kun Zhang, Spectral analysis of the transfer operator for the Lorentz gas, J. Mod. Dyn. 5 (2011), no. 4, 665–709. MR 2903754, DOI 10.3934/jmd.2011.5.665
Frédéric Faure, Nicolas Roy, and Johannes Sjöstrand, Semi-classical approach for Anosov diffeomorphisms and Ruelle resonances, Open Math. J. 1 (2008), 35–81. MR 2461513, DOI 10.2174/1874117700801010035
F. Faure and M. Tsujii, Fractal Weyl law for the Ruelle spectrum of Anosov flows, arXiv:1706.09307.
Jens Franke, On the spaces $\textbf {F}_{pq}^s$ of Triebel-Lizorkin type: pointwise multipliers and spaces on domains, Math. Nachr. 125 (1986), 29–68. MR 847350, DOI 10.1002/mana.19861250104
Sébastien Gouëzel and Carlangelo Liverani, Banach spaces adapted to Anosov systems, Ergodic Theory Dynam. Systems 26 (2006), no. 1, 189–217. MR 2201945, DOI 10.1017/S0143385705000374
Carlangelo Liverani and Dalia Terhesiu, Mixing for some non-uniformly hyperbolic systems, Ann. Henri Poincaré 17 (2016), no. 1, 179–226. MR 3437828, DOI 10.1007/s00023-015-0399-8
Thomas Runst and Winfried Sickel, Sobolev spaces of fractional order, Nemytskij operators, and nonlinear partial differential equations, De Gruyter Series in Nonlinear Analysis and Applications, vol. 3, Walter de Gruyter & Co., Berlin, 1996. MR 1419319, DOI 10.1515/9783110812411
Robert S. Strichartz, Multipliers on fractional Sobolev spaces, J. Math. Mech. 16 (1967), 1031–1060. MR 0215084
Hans Triebel, General function spaces. III. Spaces $B_{p,q}^{g(x)}$ and $F_{p,q}^{g(x)}$, $1<p<\infty$: basic properties, Anal. Math. 3 (1977), no. 3, 221–249 (English, with Russian summary). MR 628468, DOI 10.1007/BF02297695